Let (sn) be a sequence. If t is in R, then there is a subsequence of (sn) converging to t if and only if the set sn - t is ______ for all E > 0. A. Finite. B. Infinite. C. Bo…

Question

Let (sn) be a sequence.

If t is in R, then there is a subsequence of (sn) converging to t if and only if the set n exists in N: is ______ for all E > 0.
A. Finite.
B. Infinite.
C. Bounded.
D. Unbounded.

If the sequence (sn) is unbounded above, it has a subsequence with limit ____
A. Infinity.
B. Negative infinity.
C. Any real number.
D. The limit does not exist.

If (sn) is unbounded below, a subsequence has limit ____
A. Infinity.
B. Negative infinity.
C. Any real number.
D. The limit does not exist.

Answer 1

The set must be infinite for a subsequence to converge to t, unbounded above has a subsequence with limit infinity, and unbounded below has a subsequence with limit negative infinity.

Step-by-step explanation:

In order for there to be a subsequence of (sn) converging to t, the set n exists in N: must be infinite for all E > 0. This means that there are infinitely many terms in the sequence that are within a small distance E of t. The correct answer is option B: Infinite.

If the sequence (sn) is unbounded above, there will be a subsequence with limit Infinity. This means that the subsequence will approach infinity as the indices increase.

If the sequence (sn) is unbounded below, there will be a subsequence with limit Negative infinity. This means that the subsequence will approach negative infinity as the indices increase.